Optimal. Leaf size=255 \[ \frac {65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac {32768 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}-\frac {256 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{1615 a^4}+\frac {64 b^2 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{133 a^2}+\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a} \]
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Rubi [A] time = 0.42, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {2016, 2002, 2014} \[ \frac {65536 b^8 \left (a x+b x^{2/3}\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac {32768 b^7 \left (a x+b x^{2/3}\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (a x+b x^{2/3}\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (a x+b x^{2/3}\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (a x+b x^{2/3}\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}-\frac {256 b^3 \left (a x+b x^{2/3}\right )^{5/2}}{1615 a^4}+\frac {64 b^2 \sqrt [3]{x} \left (a x+b x^{2/3}\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (a x+b x^{2/3}\right )^{5/2}}{133 a^2}+\frac {2 x \left (a x+b x^{2/3}\right )^{5/2}}{7 a} \]
Antiderivative was successfully verified.
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Rule 2002
Rule 2014
Rule 2016
Rubi steps
\begin {align*} \int x \left (b x^{2/3}+a x\right )^{3/2} \, dx &=\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {(16 b) \int x^{2/3} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{21 a}\\ &=-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (32 b^2\right ) \int \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{3/2} \, dx}{57 a^2}\\ &=\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {\left (128 b^3\right ) \int \left (b x^{2/3}+a x\right )^{3/2} \, dx}{323 a^3}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (256 b^4\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{\sqrt [3]{x}} \, dx}{969 a^4}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {\left (2048 b^5\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{2/3}} \, dx}{12597 a^5}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (4096 b^6\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x} \, dx}{46189 a^6}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}-\frac {\left (16384 b^7\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{4/3}} \, dx}{415701 a^7}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}-\frac {32768 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}+\frac {\left (32768 b^8\right ) \int \frac {\left (b x^{2/3}+a x\right )^{3/2}}{x^{5/3}} \, dx}{2909907 a^8}\\ &=-\frac {256 b^3 \left (b x^{2/3}+a x\right )^{5/2}}{1615 a^4}+\frac {65536 b^8 \left (b x^{2/3}+a x\right )^{5/2}}{4849845 a^9 x^{5/3}}-\frac {32768 b^7 \left (b x^{2/3}+a x\right )^{5/2}}{969969 a^8 x^{4/3}}+\frac {8192 b^6 \left (b x^{2/3}+a x\right )^{5/2}}{138567 a^7 x}-\frac {4096 b^5 \left (b x^{2/3}+a x\right )^{5/2}}{46189 a^6 x^{2/3}}+\frac {512 b^4 \left (b x^{2/3}+a x\right )^{5/2}}{4199 a^5 \sqrt [3]{x}}+\frac {64 b^2 \sqrt [3]{x} \left (b x^{2/3}+a x\right )^{5/2}}{323 a^3}-\frac {32 b x^{2/3} \left (b x^{2/3}+a x\right )^{5/2}}{133 a^2}+\frac {2 x \left (b x^{2/3}+a x\right )^{5/2}}{7 a}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 135, normalized size = 0.53 \[ \frac {2 \left (a \sqrt [3]{x}+b\right )^2 \sqrt {a x+b x^{2/3}} \left (692835 a^8 x^{8/3}-583440 a^7 b x^{7/3}+480480 a^6 b^2 x^2-384384 a^5 b^3 x^{5/3}+295680 a^4 b^4 x^{4/3}-215040 a^3 b^5 x+143360 a^2 b^6 x^{2/3}-81920 a b^7 \sqrt [3]{x}+32768 b^8\right )}{4849845 a^9 \sqrt [3]{x}} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 602, normalized size = 2.36 \[ -\frac {2}{692835} \, b {\left (\frac {32768 \, b^{\frac {19}{2}}}{a^{9}} - \frac {\frac {19 \, {\left (6435 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} - 58344 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b + 235620 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{2} - 556920 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{3} + 850850 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{4} - 875160 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{5} + 612612 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{6} - 291720 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{7} + 109395 \, \sqrt {a x^{\frac {1}{3}} + b} b^{8}\right )} b}{a^{8}} + \frac {9 \, {\left (12155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} - 122265 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b + 554268 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{2} - 1492260 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{3} + 2645370 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{4} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{5} + 2771340 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{6} - 1662804 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{7} + 692835 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{8} - 230945 \, \sqrt {a x^{\frac {1}{3}} + b} b^{9}\right )}}{a^{8}}}{a}\right )} + \frac {2}{1616615} \, a {\left (\frac {65536 \, b^{\frac {21}{2}}}{a^{10}} + \frac {\frac {21 \, {\left (12155 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} - 122265 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b + 554268 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{2} - 1492260 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{3} + 2645370 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{4} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{5} + 2771340 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{6} - 1662804 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{7} + 692835 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{8} - 230945 \, \sqrt {a x^{\frac {1}{3}} + b} b^{9}\right )} b}{a^{9}} + \frac {5 \, {\left (46189 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {21}{2}} - 510510 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {19}{2}} b + 2567565 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {17}{2}} b^{2} - 7759752 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {15}{2}} b^{3} + 15668730 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {13}{2}} b^{4} - 22221108 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {11}{2}} b^{5} + 22632610 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} b^{6} - 16628040 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} b^{7} + 8729721 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} b^{8} - 3233230 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} b^{9} + 969969 \, \sqrt {a x^{\frac {1}{3}} + b} b^{10}\right )}}{a^{9}}}{a}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 112, normalized size = 0.44 \[ \frac {2 \left (a x +b \,x^{\frac {2}{3}}\right )^{\frac {3}{2}} \left (a \,x^{\frac {1}{3}}+b \right ) \left (692835 a^{8} x^{\frac {8}{3}}-583440 a^{7} b \,x^{\frac {7}{3}}+480480 a^{6} b^{2} x^{2}-384384 a^{5} b^{3} x^{\frac {5}{3}}+295680 a^{4} b^{4} x^{\frac {4}{3}}-215040 a^{3} b^{5} x +143360 a^{2} b^{6} x^{\frac {2}{3}}-81920 a \,b^{7} x^{\frac {1}{3}}+32768 b^{8}\right )}{4849845 a^{9} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a x + b x^{\frac {2}{3}}\right )}^{\frac {3}{2}} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x\,{\left (a\,x+b\,x^{2/3}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x \left (a x + b x^{\frac {2}{3}}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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